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weakBruhatOrder -- compares two permutations in the weak Bruhat order

Description

The (right) weak Bruhat order is a partial order on the symmetric group $\mathfrak{S}_n$. For two permutations $p$ and $q$, $w \leq_R v$ if and only if $\ell(w) + \ell(v^{-1} w) = \ell(v)$, where $\ell$ denotes the length(Permutation) of a permutation and $\leq_R$ is the right weak Bruhat order. See [BB05] for more details on the weak Bruhat order.

i1 : p = permutation {1,3,2}

o1 = Permutation{1, 3, 2}

o1 : Permutation
i2 : q = permutation {2,1,3}

o2 = Permutation{2, 1, 3}

o2 : Permutation
i3 : r = permutation {3,1,2}

o3 = Permutation{3, 1, 2}

o3 : Permutation
i4 : weakBruhatOrder(p, r)

o4 = true
i5 : weakBruhatOrder(q, r)

o5 = false

By default, weakBruhatOrder computes the right weak Bruhat order, but the optional argument {tt Side} can be used to compute, for example, the left weak Bruhat order. The current options for Side are "left" and "right".

i6 : p = permutation {1,3,2}

o6 = Permutation{1, 3, 2}

o6 : Permutation
i7 : r = permutation {3,1,2}

o7 = Permutation{3, 1, 2}

o7 : Permutation
i8 : weakBruhatOrder(inverse p, inverse r, Side=>"left")

o8 = true

References

See also

Ways to use weakBruhatOrder:

  • weakBruhatOrder(Permutation,Permutation)

For the programmer

The object weakBruhatOrder is a method function with options.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/Permutations/Documentation/mainDocs.m2:850:0.